RESEARCH ARTICLE

Modelling optimal allocation of resources in

the context of an incurable disease

Betty Nannyonga1☯*, David J. T. Sumpter2☯

1 Department of Mathematics, School of Physical Sciences College of Natural Sciences, Makerere

University, Kampala, Uganda, 2 Department of Mathematics, Center for Interdisciplinary Mathematics

Mathematics Institution, Uppsala University, Uppsala, Sweden

☯ These authors contributed equally to this work.

* bnk@cns.mak.a.ug

Abstract

Nodding syndrome has affected and led to the deaths of children between the ages of 5 and

15 in Northern Uganda since 2009. There is no reliable explanation of the disease, and cur-

rently the only treatment is through a nutritional programme of vitamins, combined with med-

ication to prevent symptoms. In the absence of a proper medical treatment, we develop a

dynamic compartmental model to plan the management of the syndrome and to curb its

effects. We use incidence data from 2012 and 2013 from Pader, Lamwo and Kitgum regions

of Uganda to parameterize the model. The model is then used to look at how to best plan the

nutritional programme in terms of first getting children on to the programme through out-

reach, and then making sure they remain on the programme, through follow-up. For the cur-

rent outbreak of nodding disease, we estimate that about half of available resources should

be put into outreach. We show how to optimize the balance between outreach and follow-up

in this particular example, and provide a general methodology for allocating resources in

similar situations. Given the uncertainty of parameter estimates in such situations, we per-

form a robustness analysis to identify the best investment strategy. Our analysis offers a

way of using available data to determine the best investment strategy of controlling nodding

syndrome.

Introduction

In the northern part of Uganda, thousands of children have fallen ill with a little known, incur-

able nodding disease syndrome, NDS. Communities are panicking and losing hope as neither

the cause nor a cure are known [1]. The affected Ugandan districts include Kitgum, Lamwo,

Pader and Gulu. The nodding syndrome emerged in Sudan in the 1960s [2], although it was

first reported in 1962 in secluded mountainous regions of Tanzania [3]. Since then, it has been

reported in small regions in South Sudan, Tanzania and Uganda [4, 5]. However, the connec-

tion between that disease and the nodding syndrome was just made recently [5].

Nodding disease affects young children between the ages of five to fifteen. It is mentally and

physically disabling, eventually fatal. As of 2013, there were more than 3,094 cases and 170

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OPENACCESS

Citation: Nannyonga B, Sumpter DJT (2017)

Modelling optimal allocation of resources in the

context of an incurable disease. PLoS ONE 12(3):

e0172401. https://doi.org/10.1371/journal.

pone.0172401

Editor: Delmiro Fernandez-Reyes, University

College London, UNITED KINGDOM

Received: July 1, 2016

Accepted: February 3, 2017

Published: March 13, 2017

Copyright: © 2017 Nannyonga, Sumpter. This is an

open access article distributed under the terms of

the Creative Commons Attribution License, which

permits unrestricted use, distribution, and

reproduction in any medium, provided the original

author and source are credited.

Data Availability Statement: All relevant data are

within the paper and its Supporting Information

files.

Funding: BN acknowledges funding from the

Faculty for the Future and the Center for

Interdisciplinary Mathematics, Mathematics

Institution, Uppsala University, Sweden. The

funders had no role in study design, data collection

and analysis, decision to publish, or preparation of

the manuscript.

Competing interests: The authors have declared

that no competing interests exist.

deaths in northern Uganda [6]. Between 2012 -2013, the government of Uganda set up out-

reach and screening centers at health facilities but due to deep poverty, many families could

not afford to pay for transport to these facilities. Children affected by NDS are characterized

by complete and permanent stunted physical growth [7]. The brain is also stunted, leading to

mental retardation of the symptomatic individual. Due to pathological bobbing of the head,

the disease is named nodding syndrome. The nodding was discovered to be an atonic seizure

[8], which often begins when children eat or feel cold. The seizures are brief and stop when the

child stops eating or feels warm again and they can manifest with a wide degree of severity.

When severe, the affected child can collapse, which may result in further injury [9], such as

falling into the open fire. Although inconclusive, there is documented association of the dis-

ease with malnutrition and onchocerciasis. NDS is a progressive fatal disorder with a duration

of about three years or more [10]. A small fraction of the symptomatic survive, and the major-

ity dies [7].

Nodding disease started spreading across the Sudan-Uganda border in 2009 where more

than 2000 cases [3] were reported in Uganda’s Kitgum district [4]. By the end of 2011, NDS

outbreaks were concentrated in Kitgum, Pader and Gulu [11]. The spread and manifestation

of outbreaks may be made worse by the poor health care of the region [10]. Children suffering

from nodding disease syndrome are exposed to poverty and malnutrition, creating a fresh dis-

tress to the already suffering region [12], and exposing children to otherwise preventable

deaths. The recent results from CDC [13] report crystals found in victims’ brains. Still, there

are no answers to why the illness attacks only children, and in most cases, some and not all in a

family. Further, there is no answer of whether there is a genetic predisposition. The impact of

NDS has been highly destructive and health officials are left mystified as to cause and cure.

Children living under the poorest conditions with insufficient food, no safe drinking water, or

indecent housing are most susceptible to this condition. In addition, the affected children are

often neglected by their parents [14] and barred from schools [15].

The outreach and screening centers set up by the Uganda government provide nourishing

foods and medication for symptoms to the patients. Each month, patients are examined to

determine any improvement in the level of malnutrition, frequency and severity of seizures.

The nutritional programme is aimed to minimize the symptoms such as occurrences of sei-

zures. However, resources are extremely limited. The research question we address is how to

balance the use of resources in getting children on to a nutritional programme and symptom

preventing medicines in the first instance through outreach, against keeping them on this

treatment programme once they have started, through follow-up. In many Uganda health con-

texts, outreach means that services that are not easily available are taken nearer to the popu-

lace. Once these services are set up, getting the children to go to the centers involves

community effort through local councils visiting their respective villages and urge the next of

kin to take them there. Follow-up is when children return the following month for check up

and additional food supplements where required. In the absence of any cure in the foreseeable

future, we look at how to best utilize the limited available resources to minimize the impact of

the disease. We address this problem using a mathematical model.

Mathematical models have been used for optimization of disease prevention [16–19] and

epidemic control [20–26]. The models enable more efficient searches for desirable control

strategies by considering all strategies simultaneously. The results direct the models in generat-

ing detailed predictions of potential future outbreaks. An optimization model on foot-and-

mouth disease [27], was used to evaluate alternative control strategies that minimized the total

regional cost for the entire epidemic duration, given disease dynamics and resource con-

straints. A similar model using optimal control theory was developed for dengue fever [28] to

analyze the optimal strategies for using these controls, and respective impact on the reduction/

Cost allocation of limited resources to control nodding syndrome

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eradication of the disease during an outbreak in the population. A model was designed to

investigate problems of disease prevention and epidemic control, in which two sets of deci-

sions namely vaccinating individuals and closing locations were optimized with the goal of

minimizing the expected number of symptomatic individuals after intervention [29]. Similar

to our objective, the goal of the paper was to formulate mathematical optimization models, to

identify which subset of individuals should be vaccinated and which locations to close in order

to minimize the expected number of symptomatic individuals. Results of such models have

proved beneficial in decision and policy making on management of disease outbreaks and

prevention.

In our model, three states are used: the symptomatic, treated and stunted. The symptomatic

children that have not been exposed to any form of treatment or food supplements. The

treated, are those that have received medication for symptoms of nodding syndrome including

malnutrition, seizures, and poor growth. The stunted are the children whose growth has been

inhibited due to nodding disease. These have either never received any medication or food

supplements, or have dropped out of treatment. Using the model, we determine the optimal

percentage of investment in treatment vs follow-up, with the overall objective of minimizing

stunted growth of the affected children.

Available data

The data used in this study was collected from Pader and Lamwo districts in the northern part

of Uganda between 2012 and 2013 by the Ministry of Health Uganda. Outreach centers were

established in the districts and serviced villages in respective parishes and sub-counties. A

team consisting of twenty (20) personnel from the Ministry of Health visited children once

every month in ten (10) different centers where each health official worked for six (6) hours

from 10:00 am to 4:00 pm, giving a total of 1200 man hours. In order to operationalize this

data to use it in the model, for each the 994 patients, we recorded

1. Age at first visit;

2. Date of initial onset;

3. How many times the patient accessed medical treatment again within six months;

4. The home villages of each of the patients.

Children were brought into the health centers by their parents or next of kin who were

informed of the study and verbal consent obtained to treat, evaluate and follow-up. On their

first evaluation, history and physical assessment of the affected children was done and

recorded. The healthcare officials used the gathered information to determine the severity of

the syndrome and level of malnutrition for each patient. In severe cases of acute malnutrition,

food supplements including ready-to-use- therapeutic foods (RUTF) were given to control

undernutrition and prevent death. Sodium valproate (INN), an anticonvulsant used in the

treatment of epilepsy, was given to control the atonic seizures. On subsequent visits, individu-

als were assessed on whether they were taking the drugs consistently or if they defaulted. The

data used in this study was authorized by the Ministry of Health Uganda who approved this

retrospective study. The records used in the study were anonymized and de-identified prior to

analysis.

Model formulation and analysis

We consider a population of children with and without nodding syndrome. Children with the

syndrome are divided into three sub groups depending on whether they seek medication and

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follow-up or not. Subgroup I gives the total number of children, who were previously healthy,

but currently with malnutrition, reported head nodding and seizure episodes. Subgroup T are

the children enrolled by the healthcare officials to receive monthly medication for displayed

symptoms and/or food supplements in case of malnutrition. Children in R have stunted

growth, classified based on visible growth impairment that was determined using the current

height as compared to previous determinations and body weight corresponding to current

age. The children are impaired due to absence of or fall-out from treatment and malnutrition.

More details are shown in Fig 1.

Nodding syndrome has been reported to exist in children aged between 5–15 years [2, 30,

31]. From the data, the minimum age at which a child was recorded to have nodding syn-

drome symptoms was 2 and the maximum 27. However, out of the 994 individuals recorded,

947 (95%) are between the ages of 5–16. Therefore, the age group of 5–16 is used to estimate

the rate at which children develop nodding syndrome. A child enters subgroup I at a rate r,

when they start to display nodding syndrome symptoms such as head bobbing, seizures and

malnutrition. Prior studies have given the prevalence of the nodding syndrome in Kitgum dis-

trict to be 12 per 1000 [30, 31]. We shall use the same value for Pader and Lamwo since they

are in the same geographical region sharing borders. In order to transform this estimate into a

rate per month we divide by the age range over which the disease can occur and the months in

a year, i.e. r = 12/(1000 × 11 × 12) = 0.00009. The rate of development of nodding syndrome is

a function of the total number of children NC, in each district. In Kitgum, 45,800 children were

reported in the age range of 5–15. The total population of Kitgum is 247,800, implying that

45,800/247,800� 18.48% is the percentage of susceptibles. We estimate NC for Lamwo and

Pader by taking 18.48% of their total populations. The population of Lamwo is 171,300, and

231,700 for Pader [32], giving the estimated number of children in these two districts as NC =

0.1848 × (171,300 + 231,700)� 74,474. Therefore, the number of incident cases that develop

nodding syndrome per month rNC = 6.7027.

The interest of this study is to minimize the number of children that develop stunted

growth. We can do this by moving children from compartment I to T, and prevent move-

ment to R. Children from subgroup I can move to subgroup T if they seek treatment, or R if

they do not seek medication for the symptoms. Movement from I to R occurs when children

develop stunted growth due to lack of medication for displayed symptoms. We assume that

on average, a child becomes removed due to NDS after six months from onset. Therefore,

the untreated development rate of nodding syndrome γ = 1/6. According to the data, of the

Fig 1. Flow diagram for the dynamics of nodding syndrome.

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994 recorded cases, 703 accessed medication and nutritional supplements at least twice in

six months. Therefore, the proportion that did not access medication at least twice in six

months is 291/994 = 0.2928. This gives the rate of fall-out from medication per month as

β = 0.2928/2 = 0.1464.

The movement from I into T depends on whether children seek medication, which in turn

depends on the outreach effort by healthcare personnel. Movement from T to R depends on

the efficiency of follow-up by medical personnel. In the absence of follow-up activities, chil-

dren can start to display nodding symptoms. Resources for outreach and follow-up are limited,

and medication for existing cases is prioritized. Professionals from the Ministry of Health set

up ten (10) health centers in each district, where they visited each center and attended to the

children once every month. The total number of hours they invested per month can be esti-

mated from the data using the number of hours each healthcare official used on each visit at

each center. Since 20 healthcare providers visited the districts for 6 hours a day (from 10:00am

to 4:00pm), 10 times per month, the total available man hours per month is H = 20 × 6 ×10 = 1200.

Of the total available man hours per month H, a proportion is used for treatment of existing

cases. We estimate the time taken to handle one child identified with nodding syndrome and

in treatment to 30 minutes [33, 34]. Thus each treatment takes τ = 30/60 = 0.5 hours. There is

one visit per treated case per month and therefore the total hours per month used for treat-

ment is τT. As a result, the total available hours for outreach or follow-up is H − τT.

Of the total remaining hours after treatment H − τT, a proportion can be invested in out-

reach and some in follow-up. Let pt be the proportion of hours dedicated to outreach activities

for getting children in treatment, and pf for the proportion of hours dedicated to follow-up,

with pt + pf = 1. Then, the total number of hours used for outreach that gets children into treat-

ment is pt(H − τT) per month, with pf(H − τT) hours used for follow-up. We do not set pt at

this point, but rather aim to find a value of pt that gives the best strategy to invest in getting

and keeping children into treatment.

In order to optimize pt and pf, we have to make modelling assumptions about how effort is

translated into success. For the transition rate from I to T the rate can be anything from 0 if no

hours are invested in outreach to 1, for a situation where healthcare is widely available, and

symptomatic children enter treatment within one month of displaying symptoms. We set a

constant α = 1 to give the saturation level of outreach when at its most effective. The transition

function of children from state I to T can then be written as

aptðH � tTÞ

Aþ ptðH � tTÞ

so that it increases with investment in outreach. τ is the time taken to handle one child per

visit, set to 30 minutes [33, 34] and A is the half saturation constant, which is the point where

children from at least 50% of the villages received medication. This value can be estimated by

taking 50% of the total number of villages in the data (211), giving A = 50% × 211 = 105.5.

For the transition rate from T to R, a child would fall-out from treatment and develop

stunted growth at a rate β. The fall-out rate can be close to 1 when more hours are invested in

getting children into treatment and close to 0 if follow-up care of at least twice in six months is

available. The constant β = 0 signifies the success and effectiveness of follow-up care. The tran-

sition function of children from state T to R is given as

bg 1 � expð� tT=ðpf ðH � tTÞÞÞ� �

;

and is large if investment in follow-up is small.

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With the descriptions of I, T and R, we can assume that δR > δI > δT since children in T are

receiving medication and follow-up and therefore die at a slower rate than I, whereas children

in R are stunted and sometimes brain dead due to the numerous seizures and more often die

faster from other accidents such as falling in open fires during abrupt seizures. With outreach,

medication and follow-up, symptomatic children recover within a period of three years

(d = 3 × 12 = 36 months) or more, or die from the symptoms [10]. With the assumption that

36 months is the time an individual should recover or die from the nodding syndrome, we set

the disease induced death rates for I to δI = 1/36, for T to δT = 0.5 × δI, and R to δR = 1.5 × δI.

These modification parameters are guesses that respectively inhibit or enhance death rates

with or without medication. We summarize all the parameters discussed in Table 1, and give a

set of equations describing the process in Eq (1).

From the above descriptions and assumptions, we can now write the system of equations

describing the system as follows:

dIdt¼ rNC �

aptðH � tTÞAþ ptðH � tTÞ

I � gI � dI I;

dTdt

¼aptðH � tTÞ

Aþ ptðH � tTÞI � bg 1 � e

� tTpf ðH � tTÞ

0

B@

1

CA � dT T;

dRdt¼ gI þ bg 1 � e

� tTpf ðH � tTÞ

0

B@

1

CA � dRR:

ð1Þ

To analyze the model in Eq (1), we start by transforming it using the following approximations:

If the proportion of investment in getting children in treatment is very small, (i.e. small pt(H −τT)), αpt(H − τT)I/(A + pt(H − τT)) can be approximated to αpt(H − τT)I/(A). Small investment

in getting children in treatment implies big investment in follow-up. Therefore, when pt(H −τT) is very small, pf(H − τT) is very big. At this point, βγ(1 − exp(τT/(pf(H − τT)))) tends to

Table 1. Parameter estimates from Uganda data.

Parameter Description Dimensions

r Development rate 0.00009 / month

NC Number of children 74,474

γ Stunted growth development rate 1/6/month

β Medication fall-out rate 0.1464 /month

H Total man hours per month 1200 man hours/month

τ Number of hours per visit per child 0.5 hours/visit/month

pt Proportion in treatment [0,1]

pf Proportion in information [0, 1 − pt]

α Rate at which treatment starts in ideal situation 1/month

A Half saturation constant 105.5

d Duration of nodding syndrome before recovery/death 36 months

δI Disease induced death rate of symptomatic 1/36 /month

δT Disease induced death rate of those on treatment 0.5/36 /month

δR Disease induced death rate of the stunted 1.5/36 /month

Parameter estimates for Lamwo and Pader.

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βγτT/(pf(H − τT)). Since pt + pf = 1, we make the substitution pf = 1 − pt and approximate

Model (1) to

dIdt¼ rNC �

aptðH � tTÞA

I � gI � dI I;

dTdt

¼aptðH � tTÞ

AI �

bgt

ð1 � ptÞðH � tTÞT � dTT;

dRdt¼ gI þ

bgt

ð1 � ptÞðH � tTÞT � dRR:

ð2Þ

In terms of the treated T�, the equilibrium points are given by

I� ¼A

aptðH � tT�Þbgt

ð1 � ptÞðH � tT�Þþ dT

� �

T�;

R� ¼1

dR

gAaptðH � tT�Þ

� �bgt

ð1 � ptÞðH � tT�Þþ dT

� �

þbgt

ð1 � ptÞðH � tT�Þ

� �

T�;

0 ¼ rNC �AT �

aptðH � tT�ÞaptðH � tT�Þ

Aþ ðgþ dIÞ

� �bgt

ð1 � ptÞðH � tT�Þþ dT

� �

:

ð3Þ

Due to complexity of the model, analysis can be done numerically to obtain mathematical con-

clusions by simulating the steady states for different values of pt 2 [0, 1]. We assume that once

the children with nodding syndrome are identified, they all opt for treatment within a month.

This assumption helps us to obtain the point for optimal investment in the benefit of treatment

and information. The optimal value of pt corresponds to the best strategy of investment and

gives the minimum number of patients that develop stunted growth or equivalently the maxi-

mum number enrolled for medication.

Fig 2 (blue line) shows that at the beginning of investment in outreach (about 2%), there is

no significant increase in the number of children enrolled in treatment. However, the red line

representing the stunted children shows a slight fall in the number of stunted growth followed

by a sudden jump, and then later to a steady decrease to a minimun value attained when opti-

mal investment is achieved. With further outreach, the blue curve shows the point at which the

number of children on treatment T increase to a maximum. This corresponds to the optimal

percentage of investment in outreach, (that is, getting children into treatment) poptt ¼ 0:51

above which further investment does not yield payoff as the number of children in treatment

begin to go down. This is due to the fact that children with nodding syndrome need regular

follow-up visits for food supplements and an anticonvulsant to control the atonic seizures.

Without follow-up, there is no evident improvement, and this leads could lead to less children

enrolling for medication and supplements. The red curve (representing stunted children)

shows the point when R is minimum obtained when pt ¼ poptt : This is the point when pmin

f ¼

1 � poptt below which investment in follow-up does not prevent children from dropping out of

treatment. It is observed from the red curve that increase in pt corresponds to a reduced pf and

results in increase in stunted growth R. These results are obtained under the assumption that

healthcare is widely available, and symptomatic children enter treatment within one month of

displaying symptoms, to give the saturation level of outreach when at its most effective.

Using the value of poptt obtained from Fig 2, the model is simulated to study the behavior of

states I, T and R around this point for pt < poptt and pt > popt

t : Fig 3 shows the change in the

number of symptomatic children I, those enrolled in treatment T and the stunted or removed

R when pt = 0.1, 0.51, and 0.9. It is observed from the figure that the point poptt is obtained after

2 months. This implies that it gets harder to keep children in treatment after two successive

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visits, and further investment without keeping those enrolled does not prevent stunted growth.

The number of children after two months of treatment with the different levels of investment

in outreach is approximately I(0.10) = 227, I(0.51) = 112, and I(0.90) = 97. We see that the

jump in the number of children on treatment from 10% to 51% investment is 115, while that

from 51% to 90% is 15. Similarly, T(0.10) = 556, T(0.51) = 716, and T(0.90) = 738; and R(0.10)

= 177, R(0.51) = 138, and R(0.90) = 132, with big jumps from T(0.10) to T(0.51) as compared

to T(0.51) to T(0.90). Thus when optimal investment in treatment is attained further input

does not yield significant benefits.

To obtain poptt , it was assumed that the treatment rate α is 1, while the rate of stunted devel-

opment after fall-out from treatment β is 0.1464. However, we can obtain different optimal

investment strategies for different sets of combinations of α and β to minimize stunted growth.

This is done by simulating the model for α, β 2 [0, 1] to determine when to invest in getting

children into treatment and when to switch to follow-up.

Fig 2. Simulation of the steady states with pt 2 [0, 1]. The red line is for the stunted while the blue line is for the children on treatment.

Note that maximum T (minimum R) is achieved at the point pt = 0.51. This gives poptt : The parameter values used are given in Table 1.

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Fig 4a and 4b shows the results of log(α) simulated against log(β) for optimal investment in

getting children into treatment pt and the respective number of children who develop stunted

growth R. It is observed from the figure that when the treatment rate α is large, it does not pay-

off to invest more in getting children in treatment and we should optimize follow-up. The

value of pt that minimizes stunted growth depends on α and β. For most of the α and β values,

pt is large indicating that investment in getting children into treatment does not prevent

stunted growth. These results are observed over a wide range of α and β although direct invest-

ment in getting children into treatment is optimal when α is small and β large.

Discussion and conclusion

Nodding syndrome has lasted a period of over five years in Pader and Lamwo districts in

northern Uganda killing mainly children between 5 and 15 years [2, 30, 31]. As a result, more

than 3,094 children are symptomatic and over 170 dead, in the war torn region with extreme

poverty. This paper provides a possible strategy of how limited resources can be utilized to

optimize control of the disease, first by getting children into treatment, and secondly, keeping

Fig 3. Simulation of the model for different values of pt 2 [0, 1], and initial conditions I = 1,000, T = 0, and R = 0. Parameter values used are in Table 1.

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them there. We hypothesize that getting children into treatment is necessary but keeping them

there is sufficient to prevent the affected children from stunted growth. We design a mathe-

matical model parameterized using data from Pader and Lamwo districts of Uganda, to pro-

vide a reliable optimal investment strategy that prevents stunted growth. The model is used to

show that if all children access treatment within a month of displaying symptoms, then up to

51% of available resources should be invested in getting children into treatment and not less

than 49% should be put in follow-up. Fig 2a shows that there is a maximum number of chil-

dren in treatment at the optimal point, suggesting that further investment would not be eco-

nomical. Fig 2b shows that as investment into medication is increased, the number of stunted

children reduce. At the point when optimal investment into getting children into treatment is

attained, we obtain the minimum number of stunted children; beyond this point, the number

of children that develop stunted growth begin to increase. This is the point when to switch

from further investment in treatment to investing into follow-up. Looking at the number of

symptomatic children after two months with no investment (pt, pf = 0,) gives 689. Without

medication, these will eventually develop stunted growth and die. If all investment is put in fol-

low-up (that is, pf = 1, pt = 0), there are 266 stunted growth which is a 61% reduction. However,

if 51% is invested in outreach (pt = 0.51, pf = 0.49), there is a reduction of 80% stunted growth.

From this we conclude that the best strategy is to invest about half in outreach, and half on fol-

low-up and predicts that the percentage number of stunted growth cases will be 80% less fol-

lowing this strategy compared to the simple strategy of all follow-up.

With the value of optimal pt obtained in Fig 2a, we simulated the model in Eq (1) to deter-

mine change if any, when values of pt < poptt and pt > popt

t are used. Fig 3a, 3b and 3c showed

the change in symptomatic, treated and stunted children with these values. Fig 3 indicates that

any investment strategy with a value of pt > poptt is not economical and does not yield any ben-

efits. This is the point at which a switch from outreach to follow-up should be made.

ure To further determine the best investment strategy, we simulated the steady states while

varying the rate α, at which treatment starts in an ideal situation, and the treatment fall-out

Fig 4. Simulation of the model for different values of α, β 2 [0, 1]. (a) Values of pt for all α and β combinations; (b) Corresponding values of R.

Parameter values used are in Table 1.

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rate β. Fig 4 postulates that when the symptomatic children access treatment immediately after

onset of symptoms and the fall-out rates are low, more investment should be put in getting

more children into treatment (Fig 4). If the children enroll in treatment immediately but the

fall-out rates are very high, then there should be little or no investment (0–20%) in getting chil-

dren into treatment; most investment (80–100%) should be utilized for follow-up. However, if

the children take very long to access treatment and the fall-out rates are very high, then up to

60% should be utilized to get more children into treatment and about 40% into follow-up.

When access to treatment is slow with low fall-outs, then all resources should be invested into

getting more children into treatment. It is important to note that the fall-out rates determined

the investment strategy for our model. Fast access to treatment coupled with high fall-out rates

call for more investment into follow-up (0–20%), while slow access to treatment with high fall-

out rates call for about 55% investment into treatment and 45% into follow-up. Fig 4 shows

that when both α and β are high, that is, fast access to treatment and high fall-out rates, it is not

optimal to further invest into getting children into treatment, since most of them are in treat-

ment. Therefore, it is more economical to invest in follow-up to reduce the high fall-out rates

and minimize the development of stunted growth.

This study provides the best way of utilizing limited resources to effectively control nod-

ding syndrome and minimize deaths and further development and stunted growth. The

results we obtained here show that prevention of stunted growth depends on two parameters;

how fast a child accesses medication α, and the probability of remaining on treatment β. In

most cases, when access to medication and fall-out rates are low, we should make greater

investment in getting more children into treatment than in follow-up, i.e optimal pt

remained high for all low values α, and more children developed stunted growth for all val-

ues of β and α< 11.85%. In most cases, optimal investment into getting children into treat-

ment pt is 100%, except for the instances when fall-out rate β is very high (>4), and access to

treatment rate α is very small (<7.29%). The results show that there is a limit to how much

you can invest in getting children in treatment and keep them there without developing

stunted growth. In the nodding case discussed here, local council representatives visited

homes with suspected children and urged their next of kins to take them to screening and

outreach centers for medication and management information. The key study limitation was

not knowing the onset of symptoms of nodding syndrome such that initial dates were taken

to be when parents suspected their children of having nodding disease after repeated seizures

and nodding. The approach we use here can help control nodding children from developing

stunted growth especially when faced with limited information and show what the benefits

of outreach and follow-up against nodding syndrome could have been. Moreover our con-

clusions are robust against the most uncertain parameters in our model such as δT and δR.

Doubling or halving these values preserves the advantage of the mixed strategy over the sin-

gle strategies. However, other diseases that have been neglected due to limited resources

could be controlled in a similar way.

Supporting information

S1 Fig.

(PDF)

S2 Fig.

(PDF)

S3 Fig.

(PDF)

Cost allocation of limited resources to control nodding syndrome

PLOS ONE | https://doi.org/10.1371/journal.pone.0172401 March 13, 2017 11 / 14

S4 Fig.

(PDF)

S1 File.

(PDF)

S2 File.

(PDF)

S3 File.

(PDF)

S4 File.

(PDF)

S5 File.

(PDF)

Acknowledgments

BN acknowledges funding from the Faculty for the Future and the Center for Interdisciplinary

Mathematics, Mathematics Institution, Uppsala University, Sweden. BN thanks the Ministry

of Health Uganda for the data. The authors extend gratitude to the constructive comments

from the reviewers.

Author Contributions

Conceptualization: BN DS.

Data curation: BN DS.

Formal analysis: BN DS.

Funding acquisition: BN DS.

Investigation: BN DS.

Methodology: BN DS.

Project administration: BN DS.

Resources: BN DS.

Software: BN DS.

Supervision: BN DS.

Validation: BN DS.

Visualization: BN DS.

Writing – original draft: BN DS.

Writing – review & editing: BN DS.

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